Group #18003537 — subgroups

Subgroups — #18003537

Order 4736.

348 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 221 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14557664	C₂	1	8	yes	maximal, complemented
2	#14557560	C₂	1	2	yes	maximal
3	#14557557	C₂	1	1	yes	maximal, characteristic
4	#14557489	C₂	1	2	yes	maximal
5	C₂×#11924447	C₂	1	1	yes	maximal, characteristic
6	C₃₇×#533	C₂	1	1	yes	maximal, characteristic
7	C₂×#10596934	C₂²	1	2	yes	—
8	C₂²×#64436	C₂²	1	1	yes	characteristic
9	C₂²×#64436	C₂²	1	1	yes	characteristic
10	C₄⋊#64436	C₂²	1	4	yes	—
11	C₄⋊#64436	C₂²	1	8	yes	—
12	#11924447	C₂²	1	4	yes	—
13	#11924447	C₂²	1	4	yes	—
14	C₄×#64436	C₂²	1	4	yes	—
15	C₄⋊Q₈×C₃₇	C₄	1	2	yes	—
16	#125×C₃₇	C₄	1	2	yes	—
17	C₂²⋊Q₈×C₃₇	C₄	1	4	yes	—
18	C₄×Q₈×C₃₇	C₂²	1	4	yes	—
19	#117×C₃₇	C₂²	1	2	yes	—
20	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₃₇	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₃₇	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₃₇	C₂×C₄	1	4	yes	—
25	C₄⋊C₄×C₃₇	C₂×C₄	1	8	yes	—
26	C₄⋊C₄×C₃₇	C₂×C₄	1	4	yes	—
27	C₂²⋊C₄×C₃₇	C₂×C₄	1	4	yes	—
28	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₃₇	C₂³	1	2	yes	—
31	C₄²×C₃₇	C₂³	1	4	yes	—
32	#10596934	—	2	4	no	—
33	C₂×#64436	—	2	2	no	—
34	C₂×#64436	Q₈	1	4	yes	complemented
35	C₂×#64436	C₂³	1	2	yes	—
36	C₂×#64436	C₂³	1	4	yes	—
37	C₂×#64436	C₂³	1	2	yes	—
38	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
39	Q₈×C₃₇	—	4	4	no	—
40	C₂×C₄×C₃₇	—	4	2	no	—
41	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
42	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
43	C₂×C₄×C₃₇	C₂²×C₄	1	8	yes	—
44	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₃₇	C₂×Q₈	1	2	yes	—
46	C₂×C₄×C₃₇	#41	1	2	yes	—
47	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
48	#64436	—	2	2	no	—
49	#64436	—	2	4	no	—
50	#64436	C₂×Q₈	1	4	yes	—
51	C₂²×C₃₇	—	2	2	no	—
52	C₂²×C₃₇	C₄×Q₈	1	2	yes	—
53	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
54	C₄×C₃₇	—	2	2	no	—
55	C₄×C₃₇	—	2	2	no	—
56	C₄×C₃₇	—	2	8	no	—
57	C₄×C₃₇	C₄×Q₈	1	2	yes	—
58	C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
59	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
60	#2961	—	37	1	no	maximal, Sylow
61	C₂×C₃₇	—	2	2	no	—
62	C₂×C₃₇	C₂×C₄×Q₈	1	1	yes	characteristic
63	C₂×C₃₇	#568	1	2	yes	—
64	#533	D₃₇	1	1	yes	characteristic
65	#446	—	37	8	no	—
66	#424	—	37	2	no	—
67	#423	—	37	1	no	—
68	#405	—	37	2	no	—
69	C₂×#97	—	37	1	no	—
70	C₃₇	#2961	1	1	yes	characteristic, complemented, Sylow
71	C₂×M₁₆	—	37	2	no	—
72	C₂²×C₈	—	37	1	no	—
73	C₂²×C₈	—	37	1	no	—
74	C₄⋊Q₈	Dic₃₇	1	2	yes	—
75	#125	Dic₃₇	1	2	yes	—
76	C₂²⋊Q₈	Dic₃₇	1	4	yes	—
77	C₄×Q₈	C₂×D₃₇	1	4	yes	—
78	#117	C₂×D₃₇	1	2	yes	—
79	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
80	C₄⋊C₈	—	37	4	no	—
81	C₄⋊C₈	—	37	8	no	—
82	#97	—	37	4	no	—
83	#97	—	37	4	no	—
84	C₄×C₈	—	37	4	no	—
85	C₂×Q₈	C₂×Dic₃₇	1	4	yes	—
86	C₂²×C₄	C₂×Dic₃₇	1	2	yes	—
87	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
88	M₁₆	—	74	4	no	—
89	C₂×C₈	—	37	4	no	—
90	C₂×C₈	—	37	2	no	—
91	C₂×C₈	—	37	4	no	—
92	C₂×C₈	—	37	2	no	—
93	C₂×C₈	—	74	2	no	—
94	C₄⋊C₄	C₂×Dic₃₇	1	4	yes	—
95	C₄⋊C₄	C₂×Dic₃₇	1	8	yes	—
96	C₄⋊C₄	C₂×Dic₃₇	1	4	yes	—
97	C₂²⋊C₄	C₂×Dic₃₇	1	4	yes	—
98	C₄²	C₂×Dic₃₇	1	1	yes	characteristic
99	C₄²	C₂×Dic₃₇	1	1	yes	characteristic
100	C₄²	C₂²×D₃₇	1	4	yes	—
101	C₄²	C₂²×D₃₇	1	2	yes	—
102	C₂³	C₂²×Dic₃₇	1	1	yes	characteristic
103	Q₈	—	4	4	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	Q₈×D₃₇	1	2	yes	—
106	C₂×C₄	#10596966	1	2	yes	—
107	C₂×C₄	C₂²×Dic₃₇	1	2	yes	—
108	C₂×C₄	C₂²×Dic₃₇	1	2	yes	—
109	C₂×C₄	C₂²×Dic₃₇	1	8	yes	—
110	C₂×C₄	C₂²×Dic₃₇	1	1	yes	characteristic
111	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
112	C₈	—	37	4	no	—
113	C₈	—	74	2	no	—
114	C₈	—	74	4	no	—
115	C₂²	—	2	2	no	—
116	C₂²	Q₈×Dic₃₇	1	2	yes	—
117	C₂²	C₂³×Dic₃₇	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	2	8	no	—
121	C₄	Q₈×Dic₃₇	1	2	yes	—
122	C₄	C₂×Q₈×D₃₇	1	1	yes	characteristic
123	C₄	C₂×#10596966	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	C₂×Q₈×Dic₃₇	1	1	yes	characteristic
126	C₂	#14558300	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14557664

C₂

yes

maximal, complemented

#14557560

C₂

yes

maximal

#14557557

C₂

yes

maximal, characteristic

#14557489

C₂

yes

maximal

C₂×#11924447

C₂

yes

maximal, characteristic

C₃₇×#533

C₂

yes

maximal, characteristic

C₂×#10596934

C₂²

yes

—

C₂²×#64436

C₂²

yes

characteristic

C₂²×#64436

C₂²

yes

characteristic

C₄⋊#64436

C₂²

yes

—

C₄⋊#64436

C₂²

yes

—

#11924447

C₂²

yes

—

#11924447

C₂²

yes

—

C₄×#64436

C₂²

yes

—

C₄⋊Q₈×C₃₇

C₄

yes

—

#125×C₃₇

C₄

yes

—

C₂²⋊Q₈×C₃₇

C₄

yes

—

C₄×Q₈×C₃₇

C₂²

yes

—

#117×C₃₇

C₂²

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

C₂×Q₈×C₃₇

C₂×C₄

yes

—

C₂²×C₄×C₃₇

C₂×C₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₄⋊C₄×C₃₇

C₂×C₄

yes

—

C₄⋊C₄×C₃₇

C₂×C₄

yes

—

C₄⋊C₄×C₃₇

C₂×C₄

yes

—

C₂²⋊C₄×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

C₂×C₄

yes

characteristic

C₄²×C₃₇

C₂×C₄

yes

characteristic

C₄²×C₃₇

C₂³

yes

—

C₄²×C₃₇

C₂³

yes

—

#10596934

—

C₂×#64436

—

C₂×#64436

Q₈

yes

complemented

C₂×#64436

C₂³

yes

—

C₂×#64436

C₂³

yes

—

C₂×#64436

C₂³

yes

—

C₂³×C₃₇

C₂²×C₄

yes

characteristic

Q₈×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂×Q₈

yes

—

C₂×C₄×C₃₇

#41

yes

—

C₂×C₄×C₃₇

C₂⁴

yes

—

—

C₂×Q₈

yes

—

C₂²×C₃₇

—

C₂²×C₃₇

C₄×Q₈

yes

—

C₂²×C₃₇

C₂³×C₄

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₄×Q₈

yes

—

C₄×C₃₇

C₂²×Q₈

yes

characteristic

C₄×C₃₇

C₂×#41

yes

characteristic

#2961

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

C₂×C₄×Q₈

yes

characteristic

C₂×C₃₇

#568

yes

—

#533

D₃₇

yes

—

—

—

—

C₂×#97

—

C₃₇

#2961

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

Dic₃₇

yes

—

#125

Dic₃₇

yes

—

C₂²⋊Q₈

Dic₃₇

yes

—

C₄×Q₈

C₂×D₃₇

yes

—

#117

C₂×D₃₇

yes

—

C₂×C₄²

C₂×D₃₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₂×Q₈

C₂×Dic₃₇

yes

—

C₂²×C₄

C₂×Dic₃₇

yes

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

C₂×Dic₃₇

yes

—

C₄⋊C₄

C₂×Dic₃₇

yes

—

C₄⋊C₄

C₂×Dic₃₇

yes

—

C₂²⋊C₄

C₂×Dic₃₇

yes

—

C₄²

C₂×Dic₃₇

yes

characteristic

C₄²

C₂×Dic₃₇

yes

characteristic

100

C₄²

C₂²×D₃₇

yes

—

101

C₄²

C₂²×D₃₇

yes

—

102

C₂³

C₂²×Dic₃₇

yes

characteristic

103

Q₈

—

104

C₂×C₄

—

105

C₂×C₄

Q₈×D₃₇

yes

—

106

C₂×C₄

#10596966

yes

—

107

C₂×C₄

C₂²×Dic₃₇

yes

—

108

C₂×C₄

C₂²×Dic₃₇

yes

—

109

C₂×C₄

C₂²×Dic₃₇

yes

—

110

C₂×C₄

C₂²×Dic₃₇

yes

characteristic

111

C₂×C₄

C₂³×D₃₇

yes

characteristic

112

C₈

—

113

C₈

—

114

C₈

—

115

C₂²

—

116

C₂²

Q₈×Dic₃₇

yes

—

117

C₂²

C₂³×Dic₃₇

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

Q₈×Dic₃₇

yes

—

122

C₄

C₂×Q₈×D₃₇

yes

characteristic

123

C₄

C₂×#10596966

yes

characteristic

124

C₂

—

125

C₂

C₂×Q₈×Dic₃₇

yes

characteristic

126

C₂

#14558300

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₇
commutator, socle	C₂²×C₃₇	C₂³×C₄
Fitting	C₃₇×#533	C₂
prime radical	C₂³×C₃₇	C₂²×C₄
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	#533	D₃₇
hypocenter	C₃₇	#2961
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ #533

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₃₇

commutator, socle

C₂²×C₃₇

C₂³×C₄

Fitting

C₃₇×#533

C₂

prime radical

C₂³×C₃₇

C₂²×C₄

cosocle

C₂×C₄×C₃₇

C₂⁴

D₃₇

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂×C₄ ◁ #533

N	H ≅ G/N	type
1. #14557664	C₂	semidirect
34. C₂×#64436	Q₈	semidirect
70. C₃₇	#2961	semidirect

H ≅ G/N

type

1. #14557664

C₂

semidirect

34. C₂×#64436

Q₈

semidirect

70. C₃₇

#2961

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#14557664	2
#14557560	2
#14557557	2
#14557489	2
C₂×#11924447	2
C₃₇×#533	2
#2961	37

subgroup

index

C₃₇×#533

Subgroups — #18003537

Conjugacy classes of subgroups

Complements

Maximal subgroups